No can be used instead of not a or not any. It is more emphatic.

I have no clothes to wear. (More emphatic than ‘I haven’t got any clothes to wear’.)
There are no letters for you. (More emphatic than ‘There aren’t any letters for you’)

None of can be used instead of not any of.

She has done none of the work I told her to do. (More emphatic than She hasn’t done any of the work I told her to do.)

After no, we usually use plural nouns unless the sense makes a singular noun necessary.

He has got no children. (More natural than ‘He has got no child’.)
I have got no clothes to wear. (More natural than ‘I have got no cloth to wear’.)

Note that when the sense is not emphatic we prefer not a or not any.

Compare:

A spider is not an insect. (NOT A spider is no insect. Here the sense is not emphatic.)
He is no fool. (More emphatic)
He is not a fool. (Less emphatic)

Not any cannot begin a sentence. Instead we use no and none of.

None of my friends wished me on my birthday. (NOT Not any of my friends wished me on my birthday.)